Write an equation for the function that is described by the given characteristics.
A cosine curve with a period of 4π,an amplitude of 7,a right phase shift of $\frac { \pi } { 2 }$ ,and a vertical translation up 2 units.

A) $y = 2 + 7 \cos \left( \frac { x } { 2 } - \frac { \pi } { 4 } \right)$
B) $y = 2 + 7 \cos \left( \frac { x } { 2 } + \pi \right)$
C) $y = 2 - 7 \cos \left( \frac { x } { 2 } - \frac { \pi } { 4 } \right)$
D) $y = 2 + 7 \cos \left( \frac { x } { 2 } - \pi \right)$
E) $y = 2 + 7 \cos ( x - \pi )$

Question

Write an equation for the function that is described by the given characteristics. ​
A cosine curve with a period of 4π,an amplitude of 7,a right phase shift of $\frac { \pi } { 2 }$ ,and a vertical translation up 2 units.
​
A) $y = 2 + 7 \cos \left( \frac { x } { 2 } - \frac { \pi } { 4 } \right)$
B) $y = 2 + 7 \cos \left( \frac { x } { 2 } + \pi \right)$
C) $y = 2 - 7 \cos \left( \frac { x } { 2 } - \frac { \pi } { 4 } \right)$
D) $y = 2 + 7 \cos \left( \frac { x } { 2 } - \pi \right)$
E) $y = 2 + 7 \cos ( x - \pi )$

Quizplus · Accepted Answer

The general form of a cosine function is $y = A \cos(B(x - C)) + D$, where $A$ is the amplitude, $B$ determines the period by $2\pi/B$, $C$ is the phase shift, and $D$ is the vertical translation. Given an amplitude of 7, a period of $4\pi$, a phase shift to the right of $\pi/2$, and a vertical translation up 2 units, we find $A = 7$, $B = 2\pi/(4\pi) = 1/2$, $C = \pi/2$, and $D = 2$. The phase shift of $\pi/2$ to the right translates to $-\pi/2$ in the argument of the cosine, which when adjusted for the value of $B$, becomes $-\pi/(2 \cdot (1/2)) = -\pi/4$. Therefore, the correct equation is $y = 2 + 7 \cos \left( \frac { x } { 2 } - \frac { \pi } { 4 } \right)$.

Write an Equation for the Function That Is Described by the Given